3.0 INTRODUCTION
This chapter provides a review of the fundamentals of static and dynamic force analy-sis, impact forces, and beam loading. The reader is assumed to have had first courses in statics and dynamics. Thus, this chapter presents only a brief, general overview of those topics but also provides more powerful solution techniques, such as the use of sin-gularity functions for beam calculations. The Newtonian solution method of force analy-sis is reviewed and a number of case-study examples are presented to reinforce understanding of this subject. The case studies also set the stage for analysis of these same systems for stress, deflection, and failure modes in later chapters.
Table 3-0 shows the variables used in this chapter and references the equations, sections, or case studies in which they are used. At the end of the chapter, a summary section is provided which groups all the significant equations from this chapter for easy reference and identifies the chapter section in which their discussion can be found.
3.1 LOADING CLASSES
The type of loading on a system can be divided into several classes based on the char-acter of the applied loads and the presence or absence of system motion. Once the gen-eral configuration of a mechanical system is defined and its kinematic motions calculated, the next task is to determine the magnitudes and directions of all the forces and couples present on the various elements. These loads may be constant or may be varying over time. The elements in the system may be stationary or moving. The most general class is that of a moving system with time-varying loads. The other combina-tions are subsets of the general class.
3
3
Table 3-1 shows the four possible classes. Class 1 is a stationary system with con-stant loads. One example of a Class 1 system is the base frame for an arbor press used in a machine shop. The base is required to support the dead weight of the arbor press which is essentially constant over time, and the base frame does not move. The parts brought to the arbor press (to have something pressed into them) temporarily add their weight to the load on the base, but this is usually a small percentage of the dead weight.
A static load analysis is all that is necessary for a Class 1 system.
Title-page photograph courtesy of Chevrolet Division of General Motors Co., Detroit, Mich.
distance to load in m Sect. 3.9
a
distance to load in m Sect. 3.9
b
damping lb-sec/in N-sec/m Eq. 3.6
d
energy in-lb joules Eq. 3.9, 3.10
E
force or load lb N Sect. 3.3
F
damped natural frequency Hz Hz Eq. 3.7
fd
natural frequency Hz Hz Eq. 3.4
fn
gravitational acceleration in/sec2 m/sec2 Eq. 3.12 g
mass moment of inertia about x axis lb-in-sec2 kg-m2 Sect. 3.3 Ix
mass moment of inertia about y axis lb-in-sec2 kg-m2 Sect. 3.3 Iy
kg-m2
mass moment of inertia about z axis lb-in-sec2 Sect. 3.3 Iz
spring rate or spring constant lb/in N/m Eq. 3.5
k
length in m Sect. 3.9
l
mass lb-sec2/in kg Sect. 3.3
m
moment, moment function lb-in N-m Sect. 3.3, 3.9
M
normal force in m Case 4A
N
beam loading function lb N Sect. 3.9
q
position vector in m Sect. 3.4
R
reaction force lb N Sect. 3.9
R
linear velocity in/sec m/sec Eq. 3.10
v
beam shear function lb N Sect. 3.9
V
weight lb N Eq. 3.14
W
generalized length variable in m Sect. 3.9
x
displacement in m Eq. 3.5, 3.8
y
deflection in m Eq. 3.5
δ
coefficient of friction none none Case 4A
μ
rotational or angular velocity rad/sec rad/sec Case 5A ω
damped natural frequency rad/sec rad/sec Eq. 3.7
ωd
natural frequency rad/sec rad/sec Eq. 3.4
ωn
correction factor none none Eq. 3.10
η
Table 3-0 Variables Used in This Chapter
Symbol Variable ips units SI units See
3 Class 2 describes a stationary system with time-varying loads. An example is a
bridge which, though essentially stationary, is subjected to changing loads as vehicles drive over it and wind impinges on its structure. Class 3 defines a moving system with constant loads. Even though the applied external loads may be constant, any signifi-cant accelerations of the moving members can create time-varying reaction forces. An example might be a powered rotary lawn mower. Except for the case of mowing the occasional rock, the blades experience a nearly constant external load from mowing the grass. However, the accelerations of the spinning blades can create high loads at their fastenings. A dynamic load analysis is necessary for Classes 2 and 3.
Note however that, if the motions of a Class 3 system are so slow as to generate negligible accelerations on its members, it could qualify as a Class 1 system and then would be called quasi-static. An automobile scissors jack (see Figure 3-5, p. 88) can be considered to be a Class 1 system since the external load (when used) is essentially constant, and the motions of the links are slow with negligible accelerations. The only complexity introduced by the motions of the elements in this example is that of deter-mining in which position the internal loads on the jack’s elements will be maximal, since they vary as the jack is raised, despite the essentially constant external load.
Class 4 describes the general case of a rapidly moving system subjected to time-varying loads. Note that even if the applied external loads are essentially constant in a given case, the dynamic loads developed on the elements from their accelerations will still vary with time. Most machinery, especially if powered by a motor or engine, will be in Class 4. An example of such a system is the engine in your car. The internal parts (crankshaft, connecting rods, pistons, etc.) are subjected to time-varying loads from the gasoline explosions, and also experience time-varying inertial loads from their own accelerations. A dynamic load analysis is necessary for Class 4.
3.2 FREE-BODY DIAGRAMS
In order to correctly identify all potential forces and moments on a system, it is neces-sary to draw accurate free-body diagrams (FBDs) of each member of the system. These FBDs should show a general shape of the part and display all the forces and moments that are acting on it. There may be external forces and moments applied to the part from outside the system, and there will be interconnection forces and/or moments where each part joins or contacts adjacent parts in the assembly or system.
In addition to the known and unknown forces and couples shown on the FBD, the dimensions and angles of the elements in the system are defined with respect to local coordinate systems located at the centers of gravity (CG) of each element.* For a dy-namic load analysis, the kinematic accelerations, both angular and linear (at the CG), need to be known or calculated for each element prior to doing the load analysis.
Table 3-1 Load Classes
Constant Loads Time-Varying Loads
Stationary Elements Class 1 Class 2
Moving Elements Class 3 Class 4
* While it is not a requirement that the local coordinate system for each element be located at its CG, this approach provides consistency and simplifies the dynamic calculations. Further, most solid modeling CAD/CAE systems will automatically calculate the mass properties of parts with respect to their CGs.
The approach taken here is to apply a consistent method that works for both static and dynamic problems and that is also amenable to computer solution.
3
3.3 LOAD ANALYSIS
This section presents a brief review of Newton’s laws and Euler’s equations as applied to dynamically loaded and statically loaded systems in both 3-D and 2-D. The method of solution presented here may be somewhat different than that used in your previous statics and dynamics courses. The approach taken here in setting up the equations for force and moment analysis is designed to facilitate computer programming of the so-lution.
This approach assumes all unknown forces and moments on the system to be posi-tive in sign, regardless of what one’s intuition or an inspection of the free-body diagram might indicate as to their probable directions. However, all known force components are given their proper signs to define their directions. The simultaneous solution of the set of equations that results will cause all the unknown components to have the proper signs when the solution is complete. This is ultimately a simpler approach than the one often taught in statics and dynamics courses which requires that the student assume di-rections for all unknown forces and moments (a practice that does help the student de-velop some intuition, however). Even with that traditional approach, an incorrect assumption of direction results in a sign reversal on that component in the solution. As-suming all unknown forces and moments to be positive allows the resulting computer program to be simpler than would otherwise be the case. The simultaneous equation solution method used is extremely simple in concept, though it requires the aid of a com-puter to solve. Software is provided with the text to solve the simultaneous equations.
See program MATRIX on the CD-ROM.
Real dynamic systems are three dimensional and thus must be analyzed as such.
However, many 3-D systems can be analyzed by simpler 2-D methods. Accordingly, we will investigate both approaches.
Three-Dimensional Analysis
Since three of the four cases potentially require dynamic load analysis, and because a static force analysis is really just a variation on the dynamic analysis, it makes sense to start with the dynamic case. Dynamic load analysis can be done by any of several meth-ods, but the one that gives the most information about internal forces is the Newtonian approach based on Newton’s laws.
NEWTON’S FIRST LAW A body at rest tends to remain at rest and a body in mo-tion at constant velocity will tend to maintain that velocity unless acted upon by an ex-ternal force.
NEWTON’S SECOND LAW The time rate of change of momentum of a body is equal to the magnitude of the applied force and acts in the direction of the force.
Newton’s second law can be written for a rigid body in two forms, one for linear forces and one for moments or torques:
F a M
∑
=m∑
G =H˙G ( .3 1a)where F = force, m = mass, a = acceleration, MG = moment about the center of grav-ity, and H˙G = the time rate of change of the moment of momentum, or the angular
mo-3 mentum about the CG. The left sides of these equations respectively sum all the forces
and moments that act on the body, whether from known applied forces or from inter-connections with adjacent bodies in the system.
For a three-dimensional system of connected rigid bodies, this vector equation for the linear forces can be written as three scalar equations involving orthogonal compo-nents taken along a local x, y, z axis system with its origin at the CG of the body:
Fx max Fy may Fz maz b
∑
=∑
=∑
= ( .3 1 )If the x, y, z axes are chosen coincident with the principal axes of inertia of the body,* the angular momentum of the body is defined as
HG =Ixωxˆi+Iyωyˆj+Izωzkˆ ( . )3 1c where Ix, Iy, and Iz are the principal centroidal mass moments of inertia (second moments of mass) about the principal axes. This vector equation can be substituted into equa-tion 3.1a to yield the three scalar equaequa-tions known as Euler’s equaequa-tions:
M I I I
M I I I d
M I I I
x x x y z y z
y y y z x z x
z z z x y x y
∑ ∑
∑
= −
(
−)
= −
(
−)
= −
(
−)
α ω ω
α ω ω
α ω ω
( .3 1 )
where Mx, My, Mz are moments about those axes and αx, αy, αz are the angular accel-erations about the axes. This assumes that the inertia terms remain constant with time, i.e., the mass distribution about the axes is constant.
NEWTON’S THIRD LAW states that when two particles interact, a pair of equal and opposite reaction forces will exist at their contact point. This force pair will have the same magnitude and act along the same direction line, but have opposite sense.
We will need to apply this relationship as well as applying the second law in order to solve for the forces on assemblies of elements that act upon one another. The six equations in equations 3.1b and 3.1d can be written for each rigid body in a 3-D sys-tem. In addition, as many (third-law) reaction force equations as are necessary will be written and the resulting set of equations solved simultaneously for the forces and mo-ments. The number of second-law equations will be up to six times the number of in-dividual parts in a three-dimensional system (plus the reaction equations), meaning that even simple systems result in large sets of simultaneous equations. A computer is needed to solve these equations, though high-end pocket calculators will solve large sets of si-multaneous equations also. The reaction (third-law) equations are often substituted into the second-law equations to reduce the total number of equations to be solved simulta-neously.
Two-Dimensional Analysis
All real machines exist in three dimensions but many three-dimensional systems can be analyzed two dimensionally if their motions exist only in one plane or in parallel planes.
* This is a convenient choice for symmetric bodies but may be less convenient for other shapes. See F. P. Beer and E. R. Johnson, Vector Mechanics for Engineers, 3rd ed., 1977, McGraw-Hill, New York, Chap. 18, “Kinetics of Rigid Bodies in Three Dimensions.”
3
Euler’s equations 3.1d show that if the rotational motions (ω, α) and applied moments or couples exist about only one axis (say the z axis), then that set of three equations re-duces to one equation,
Mz Iz z a
∑
= α ( .3 2 )because the ω and α terms about the x and y axes are now zero. Equation 3.1b is re-duced to
Fx max Fy may b
∑
=∑
= ( .3 2 )Equations 3.2 can be written for all the connected bodies in a two-dimensional system and the entire set solved simultaneously for forces and moments. The number of sec-ond-law equations will now be up to three times the number of elements in the system plus the necessary reaction equations at connecting points, again resulting in large sys-tems of equations for even simple syssys-tems. Note that even though all motion is about one (z) axis in a 2-D system, there may still be loading components in the z direction due to external forces or couples.
Static Load Analysis
The difference between a dynamic loading situation and a static one is the presence or absence of accelerations. If the accelerations in equations 3.1 and 3.2 are all zero, then for the three-dimensional case these equations reduce to
F F F
a
M M M
x y z
x y z
∑ ∑ ∑
∑ ∑ ∑
= = =
= = =
0 0 0
3 3
0 0 0
( . )
and for the two-dimensional case,
Fx Fy Mz b
∑
=0∑
=0∑
=0 ( .3 3 )Thus, we can see that the static loading situation is just a special case of the dynamic loading one, in which the accelerations happen to be zero. A solution approach based on the dynamic case will then also satisfy the static one with appropriate substitutions of zero values for the absent accelerations.
3.4 TWO-DIMENSIONAL, STATIC LOADING CASE STUDIES
This section presents a series of three case studies of increasing complexity, all limited to two-dimensional static loading situations. A bicycle handbrake lever, a crimping tool, and a scissors jack are the systems analyzed. These case studies provide examples of the simplest form of force analysis, having no significant accelerations and having forces acting in only two dimensions.
3